%I A005320 M2919
%S A005320 0,3,12,45,168,627,2340,8733,32592,121635,453948,1694157,6322680,
%T A005320 23596563,88063572,328657725,1226567328,4577611587,17083879020,
%U A005320 63757904493,237947738952,888033051315,3314184466308,12368704813917
%N A005320 a(n) = 4a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.
%C A005320 For n > 1, a(n-1) is the determinant of the n-by-n band matrix which 
               has {2,4,4,...,4,4,2} on the diagonal and a 1 on the entire super- 
               and subdiagonal. This matrix appears when constructing a natural 
               cubic spline interpolating n equally spaced data points. - g.degroot(AT)phys.uu.nl, 
               Feb 14 2007
%C A005320 Integer values of x that make Sqrt[9+3x^2] a perfect square. - Lorenz 
               H. Menke, Jr. (lnz2004(AT)mindspring.com), Mar 26 2008
%C A005320 The intermediate convergents to 3^(1/2), beginning with 3/2, 12/7, 45/
               26, 168/97, comprise a strictly increasing sequence; numerators=A005320, 
               denominators=A001075. - Clark Kimberling (ck6(AT)evansville.edu), 
               Aug 27 2008
%D A005320 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005320 E. K. Lloyd "The standard deviation of 1, 2, .., n, Pell's equation and 
               rational triangles", preprint.
%D A005320 Clark Kimberling, "Best lower and upper approximates to irrational numbers,
               " Elemente der Mathematik, 52 (1997) 122-126.
%D A005320 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, 
               New York, 1966.
%H A005320 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A005320 C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/
               poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, 
               Melbourne, 2002.
%H A005320 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A005320 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005320 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005320 E. Keith Lloyd, <a href="http://www.jstor.org/stable/3619201">The Standard 
               Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles</
               a>, Math. Gaz. vol 81 (1997), 231-243. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Apr 21 2009]
%F A005320 a(n) = 5[a(n-1)-a(n-2)] + a(n-3); a(0) = 0, a(1) = 3, a(2) = 12; n > 
               3; a(n) = (sqrt(3)/2)*(2+sqrt(3))^n-(sqrt(3)/2)*(2-sqrt(3))^n. - 
               Antonio Alberto Olivares (tonioolivares(AT)todito.com), Jan 17 2004
%p A005320 A005320:=3*z/(1-4*z+z**2); [S. Plouffe in his 1992 dissertation.]
%p A005320 a := n -> (Matrix([[3,0]]). Matrix([[4,1],[ -1,0]])^n)[1,2]; seq (a(n), 
               n=0..50); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 14 
               2008]
%t A005320 Det[SparseArray[{{i_, i_} -> If[i == 1 || i == n, 2, 4], {i_, j_} -> 
               If[Abs[i - j] == 1, 1, 0]}, {n, n}]] (* the recurrence relation is 
               faster! *) - g.degroot(AT)phys.uu.nl, Feb 14 2007
%t A005320 Do[If[IntegerQ[Sqrt[(9 + 3 x^2)]], Print[{x, Sqrt[(9 + 3 x^2)]}]], {x, 
               0, 2000000}] - Lorenz H. Menke, Jr. (lnz2004(AT)mindspring.com), 
               Mar 26 2008
%Y A005320 Cf. A082841.
%Y A005320 Sequence in context: A109437 A005656 A064017 this_sequence A062561 A128593 
               A085481
%Y A005320 Adjacent sequences: A005317 A005318 A005319 this_sequence A005321 A005322 
               A005323
%K A005320 nonn,easy
%O A005320 0,2
%A A005320 N. J. A. Sloane (njas(AT)research.att.com).
%E A005320 More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 14 2008
%E A005320 Typo in definition corrected by Johannes Boot, Feb 05 2009